Computer Science
LESSON ON
Number Base Addition
John Owen, Rockport Fulton HS
1
Objective


In this lesson you’ll learn how to do
simple addition in Base 2, 8, and
16.
It is essentially the same as Base
10 addition, just in a different base!
John Owen, Rockport Fulton HS
2
Review Base Ten Addition, #1



In Base 10 addition, you learned a
very simple process.
Look at this problem:
12
+37
First add the ones column, then the
tens.
John Owen, Rockport Fulton HS
3
Review Base Ten Addition, #1
12
+37
49

The answer is 49…simple, right?
John Owen, Rockport Fulton HS
4
Review Base Ten Addition, #2


Now look at this problem:
13
+37
When you add the ones column
values, the result of 10 EQUALS the
base value of 10, so you have to
CARRY a 1.
John Owen, Rockport Fulton HS
5
Review Base Ten Addition, #2

1
13
+37
0
When a carry is made, you essentially
divide by 10 (the base) to determine
what value to carry, and mod by 10 to
determine what value to leave behind.
John Owen, Rockport Fulton HS
6
Review Base Ten Addition,#2



1
13
+37
0
3 plus 7 is 10
10 divided by 10 is 1 (carry)
10 mod 10 is 0 (leave)
John Owen, Rockport Fulton HS
7
Review Base Ten Addition, #2
1
13
+37
50

Answer is 50
John Owen, Rockport Fulton HS
8
Review Base Ten Addition, #3


Here’s a third example:
16
+37
When you add the ones column
values, the result of 13 EXCEEDS
the base value of 10, so CARRY a
1.
John Owen, Rockport Fulton HS
9
Review Base Ten Addition, #3
16
+37



6 plus 7 is 13
13 divided by 10 is 1 (carry)
13 mod 10 is 3 (leave)
John Owen, Rockport Fulton HS
10
Review Base Ten Addition, #3

1
16
+37
53
Answer is 53
John Owen, Rockport Fulton HS
11
Review Base Ten Addition, #4


And finally, a fourth example:
76
+35
The ones column result of 11
EXCEEDS the base value of 10, and
you CARRY a 1.
John Owen, Rockport Fulton HS
12
Review Base Ten Addition,#4



1
76
+35
1
6 plus 5 is 11
11 divided by 10 is 1 (carry)
11 mod 10 is 1 (leave)
John Owen, Rockport Fulton HS
13
Review Base Ten Addition, #4



1
76
+35
1
1+7+3 is 6 plus 5, which equals 11
11 divided by 10 is 1 (carry)
11 mod 10 is 1 (leave)
John Owen, Rockport Fulton HS
14
Review Base Ten Addition, #4
1

76
+35
111
Answer is 111, base 10
John Owen, Rockport Fulton HS
15
Base Eight Addition, #1


Now here is an example in base
eight:
12
+34
When you add the ones column
values, the answer is 6, and the
second column answer is 4.
John Owen, Rockport Fulton HS
16
Base Eight Addition. #1



12
+34
48
Answer is 48, base eight
You say, “four eight base eight”, not
“forty-eight”
The phrase “forty-eight” is meant for
base ten only.
John Owen, Rockport Fulton HS
17
Base Eight Addition, #2


Now look at this problem:
14
+34
When you add the ones column
values, the result of 8 EQUALS the
base value of 8, and you have to
CARRY a one.
John Owen, Rockport Fulton HS
18
Base Eight Addition, #2
14
+34

Again you divide by 8 (the base) to
determine what value to carry, and
mod by 8 to determine what value
to leave behind.
John Owen, Rockport Fulton HS
19
Base Eight Addition, #2



1
14
+34
0
4 plus 4 is 8
8 divided by 8 is 1 (carry)
8 mod 8 is 0 (leave)
John Owen, Rockport Fulton HS
20
Base Eight Addition, #2


1
14
+34
50
Answer is “five zero, base eight”!
Looks strange, but it is correct!
John Owen, Rockport Fulton HS
21
Base Eight Addition, #3


Here’s a third example:
16
+37
When you add the ones column
values, the result of 13 EXCEEDS
the base value of 8, and you have
to CARRY a one.
John Owen, Rockport Fulton HS
22
Base Eight Addition, #3



1
16
+37
5
6 plus 7 is 13
13 divided by 8 is 1 (carry)
13 mod 8 is 5 (leave)
John Owen, Rockport Fulton HS
23
Base Eight Addition, #3

1
16
+37
55
Answer is 55, base eight.
John Owen, Rockport Fulton HS
24
Base Eight Addition, #4


And a fourth example:
76
+35
The ones column result of 11
EXCEEDS the base value of 8,
…CARRY a one.
John Owen, Rockport Fulton HS
25
Base Eight Addition, #4



1
76
+35
3
6 plus 5 is 11
11 divided by 8 is 1 (carry)
11 mod 8 is 3 (leave)
John Owen, Rockport Fulton HS
26
Base Eight Addition, #4



1
76
+35
33
1+7+3 is 6 plus 5 is 11
11 divided by 8 is 1 (carry)
11 mod 8 is 3 (leave)
John Owen, Rockport Fulton HS
27
Base Eight Addition, #4
1
76
+35
133
Answer is 133, base 8
John Owen, Rockport Fulton HS
28
Base Two Addition, #1


Base Two Addition is quite
interesting, but also fairly simple.
Since the only counting digits in
base two are the values 0 and 1,
there are only a few situations you
have to learn.
John Owen, Rockport Fulton HS
29
Base Two Addition, #1


We’ll start simple:
1
+1
=10 (“one zero, base two”)
This looks strange, but the same
process applies.
John Owen, Rockport Fulton HS
30
Base Two Addition, #1

1
+1
= 10
Since 1 + 1 is 2, this EQUALS the
base value of 2, which means you
carry the “div” answer and leave
the “mod” answer
John Owen, Rockport Fulton HS
31
Base Two Addition, #1



1
+1
= 10
2 / 2 = 1 (carry)
2 % 2 = 0 (leave)
That’s it!
John Owen, Rockport Fulton HS
32
Base Two Addition, #2


Here’s another:
10
+11
= 101
Can you figure it out?
John Owen, Rockport Fulton HS
33
Base Two Addition, #2


10
+11
= 101
In the ones column, 1 + 0 is 1.
In the second column, 1+1 is 2, or
10 in base 2
John Owen, Rockport Fulton HS
34
Base Two Addition, #3


And another:
101101
+110011
=
Can you figure it out?
John Owen, Rockport Fulton HS
35
Base Two Addition, #3

Step by step…
1
101101
+110011
=
0
John Owen, Rockport Fulton HS
36
Base Two Addition, #3

Step by step…
1
101101
+110011
=
00
John Owen, Rockport Fulton HS
37
Base Two Addition, #3

Step by step…
1
101101
+110011
=
000
John Owen, Rockport Fulton HS
38
Base Two Addition, #3

Step by step…
1
101101
+110011
= 0000
John Owen, Rockport Fulton HS
39
Base Two Addition, #3


Step by step…
1
101101
+110011
= 00000
Since 1+1+1 is 3, carry 1 and
leave 1
John Owen, Rockport Fulton HS
40
Base Two Addition, #3


Step by step…
1
101101
+110011
=1100000
All done!
John Owen, Rockport Fulton HS
41
Base Sixteen, Example #1


In base sixteen, remember the
digits are 0-9, then A-F,
representing the values 0-15
Here’s an example:
29
+12
John Owen, Rockport Fulton HS
42
Base Sixteen, Example #1


29
+12
= 3B, base 16
2 + 9 is 11, which is B in base
sixteen
2+1 is 3, so the answer is 3B
John Owen, Rockport Fulton HS
43
Base Sixteen, Example #2


1
A9
+47
= F0, base 16
9+7 is 16, equal to the base, so
carry 1 and leave 0
1 + A(10) + 4 is 15, which is F
John Owen, Rockport Fulton HS
44
Base Sixteen, Example #3


11
D6
+7C
= 152, base 16
6+C(12) = 18, carry 1, leave 2
1+D(13)+7 = 21, carry 1, leave 5
John Owen, Rockport Fulton HS
45
Base Sixteen, Example #4


11
EF
+2D
= 11C, base 16
F(15) + D(13) = 28, carry 1, leave
C(12)
1 + E(14) + 2 = 17, carry 1, leave 1
John Owen, Rockport Fulton HS
46
Exercises

1.
2.
3.
4.
5.
6.
Now try these exercises
12 + 12 =
78 + 68 =
F16 + F16 =
58 + 58 =
916 + B16 =
C16 + D16 =
John Owen, Rockport Fulton HS
47
Exercises
7.
8.
9.
10.
11.
12.
13.
38 + 4 8 =
F16 + 216 =
102 + 102 =
12 + 10112 =
102 + 1102 =
2168 + 3648 =
7778 + 38 =
John Owen, Rockport Fulton HS
48
Exercises
14.
15.
16.
ACE16 + BAD16 =
23416 + 97516 =
4216 + F16 + 87616 =
John Owen, Rockport Fulton HS
49
ANSWERS (JUMBLED)
7
11 12 10 14 15 19
1E 100
1002
602
1100
BA9 8C7
1000
167B
John Owen, Rockport Fulton HS
50